Notice that, in the setting of this definition, the set \(\lbrace g_{s+1},\dots ,g_r\rbrace \)  is the reduced \(\sigma \) -Gröbner basis of \(I\cap K[Y]\) 
by Proposition REF .d. Thus the map \(\Gamma _Z\)  is well-defined.
Our main result in this section is that the map \(\Gamma _Z\)  is bijective.
For its proof we need the following result which is part of the
theory of Gröbner fans (see [1]}). For the convenience of the readers,
we include it here together with a proof.
